Found problems: 85335
Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?
$\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\displaystyle\frac{4}{3}\qquad\textbf{(D)}~\displaystyle\frac{3}{2}\qquad\textbf{(E)}~\displaystyle\frac{5}{3}$
Let $A$ be the locus of points $(\alpha, \beta, \gamma)$ in the $\alpha\beta\gamma$-coordinate space that satisfy the following properties:
[b](I)[/b] We have $\alpha$, $\beta$, $\gamma > 0$.
[b](II)[/b] We have $\alpha + \beta + \gamma = \pi$.
[b](III)[/b] The intersection of the three cylinders in the $xyz$-coordinate space given by the equations
\begin{eqnarray*}
y^2 + z^2 & = & \sin^2 \alpha \\
z^2 + x^2 & = & \sin^2 \beta \\
x^2 + y^2 & = & \sin^2 \gamma
\end{eqnarray*}
is nonempty.
Determine the area of $A$.
Find all positive integers $n$ for which $$S(n^2)+S(n)^2=n$$ where $S(m)$ denotes the sum of digits of $m$.
Let $a$ and $b$ be two real numbers and let $n$ be a nonnegative integer. Then prove that
$$\sum_{k=0}^{n} {n \choose k} (a + k)^k (b - k)^{n-k} = \sum_{k=0}^{n} \frac{n!}{t!} (a + b)^t $$
To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals:
[img]https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png[/img]
An [i]integer-ratio[/i] right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides $3,4,5$ is an[i] integer-ratio[/i] right triangle, and so is a triangle with sides $\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3$. On the other hand, the right triangle with sides$ \sqrt2 ,\sqrt5, \sqrt7$ is not an [i]integer-ratio[/i] right triangle. Determine, with proof, all integers $n$ for which it is possible to completely [i]dissect [/i] a regular $n$-sided polygon into [i]integer-ratio[/i] right triangles.
Let $ABC$ be a triangle. On the sides $BC$, $CA$, $AB$ of the triangle, construct outwardly three squares with centres $O_a$, $O_b$, $O_c$ respectively. Let $\omega$ be the circumcircle of $\vartriangle O_aO_bO_c$. Given that $A$ lies on $\omega$, prove that the centre of $\omega$ lies on the perimeter of $\vartriangle ABC$.
[i]Sam Bealing, United Kingdom[/i]
Find all pairs $ (x; y) $ of positive integers such that
$$xy | x^2 + 2y -1.$$
A $k \times k$ array contains each of the numbers $1, 2, \dots, m$ exactly once, with the remaining entries all zero. Suppose that all the row sums and column sums are equal. What is the smallest possible value of $m$ if $k = 3^n$ ($n \in \mathbb{N}^+$)?
[i]Attila Sztranyák and Péter Erben[/i]
For positive real numbers $a,b,c$, $\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}$ is true. Prove that:
$$ \frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2$$
Let $P$ be a point inside a square $ABCD,$ such that $\angle BPC = 135^\circ $ and the area of triangle $ADP$ is twice as much as the area of triangle $PCD.$ Find $\frac {AP}{DP}.$
[i]Proposed by Andria Gvaramia, Georgia [/i]
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
Let $x=2\sin8^\circ+2\sin16^\circ+\dots+2\sin176^\circ$. What is $\arctan(x)$?
Denote by $d(n)$ the number of positive divisors of a positive integer $n$.
Prove that there are infinitely many positive integers $n$ such that $\left\lfloor\sqrt{3}\cdot d(n)\right\rfloor$ divides $n$.
When simplified, $\log{8} \div \log{\frac{1}{8}}$ becomes:
${{{ \textbf{(A)}\ 6\log{2} \qquad\textbf{(B)}\ \log{2} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 0}\qquad\textbf{(E)}\ -1}} $
Let $a$ be [i]good[/i] if the number of prime divisors of $a$ is equal to $2$. Do there exist $18$ consecutive good natural numbers?
Let $a,b,c,d \in \mathbb{R}^+$ and suppose that all roots of the equation
\begin{align*} x^5-ax^4+bx^3-cx^2+dx=1 \end{align*}
are real. Prove
\begin{align*} \frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} \le \frac{3}{5} \end{align*}
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R}\longrightarrow \mathbb{R}$ so that the equality $$f(x+yf(x+y)) +xf(x)= f(xf(x+y+1))+y^2$$
is true for any real numbers $x,y$.
Determine all three-digit numbers N such that $N^2$ has six digits and so that the sum of the number formed by the first three digits of $N^2$ and the number formed by the latter three digits of $N^2$ equals $N$.
Let $I$ be the incenter of $\triangle ABC$ and $P$ be a point such that $PI$ is perpendicular to $BC$ and $PA$ is parallel to $BC$. Let the line parallel to $BC$, which is tangent to the incircle of $\triangle ABC$, intersect $AB$ and $AC$ at points $Q$ and $R$ respectively. Prove that $\angle BPQ = \angle CPR$.
Square $ AIME$ has sides of length $ 10$ units. Isosceles triangle $ GEM$ has base $ EM$, and the area common to triangle $ GEM$ and square $ AIME$ is $ 80$ square units. Find the length of the altitude to $ EM$ in $ \triangle GEM$.
In each square of a garden shaped like a $2022 \times 2022$ board, there is initially a tree of height $0$. A gardener and a lumberjack alternate turns playing the following game, with the gardener taking the first turn:
[list]
[*] The gardener chooses a square in the garden. Each tree on that square and all the surrounding squares (of which there are at most eight) then becomes one unit taller.
[*] The lumberjack then chooses four different squares on the board. Each tree of positive height on those squares then becomes one unit shorter.
[/list]
We say that a tree is [i]majestic[/i] if its height is at least $10^6$. Determine the largest $K$ such that the gardener can ensure there are eventually $K$ majestic trees on the board, no matter how the lumberjack plays.
In triangle $ABC$, point $D$ divides side $\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $BC$ and line $AE$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$?
[asy]
unitsize(1.5cm);
pair A,B,C,DD,EE,FF;
B = (0,0); C = (3,0);
A = (1.2,1.7);
DD = (2/3)*A+(1/3)*C;
EE = (B+DD)/2;
FF = intersectionpoint(B--C,A--A+2*(EE-A));
draw(A--B--C--cycle);
draw(A--FF);
draw(B--DD);dot(A);
label("$A$",A,N);
dot(B);
label("$B$",
B,SW);dot(C);
label("$C$",C,SE);
dot(DD);
label("$D$",DD,NE);
dot(EE);
label("$E$",EE,NW);
dot(FF);
label("$F$",FF,S);
[/asy]
$\textbf{(A) }24\qquad\textbf{(B) }30\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad\textbf{(E) }40$
The functions $f,g:[0,1]\to\mathbb R$ are given with the formulas
$$f(x)=\sqrt[4]{1-x},\enspace g(x)=f(f(x)),$$
and $c$ denotes any solution of $x=f(x)$.
(a) i. Analyze the function $f(x)$ and draw its graph. Prove that the equation $f(x)=x$ has the unique root $c$ satisfying $c\in[0.72,0.73]$.
ii. Analyze the function $f'(x)$. Let $M_1$ and $M_2$ be the points of the graph of $f(x)$ with different $x$ coordinates. What is the position of the arc $M_1M_2$ of the graph with respect to the segment $M_1M_2$?
iii. Analyze the function $g(x)$ and draw its graph. What is the position of that graph with respect to the line $y=x$? Find the tangents to the graph at points with $x$ coordinates $0$ and $1$.
iv. Prove that every sequence $\{a_n\}$ with the conditions $a_1\in(0,1)$ and
$a_{n+1}=f(a_n)$ for $n\in\mathbb N$ converges.
[hide=Official Hint]Consider the sequences $\{a_{2n-1}\},\{a_{2n}\}~(n\in\mathbb N)$ and the function $g(x)$ associated with the graph.[/hide]
(b) On the graph of the function $f(x)$ consider the points $M$ and $M'$ with $x$ coordinates $x$ and $f(x)$, where $x\ne c$.
i. Prove that the line $MM'$ intersects with the line $y=x$ at the point with $x$ coordinate
$$h(x)=x-\frac{(f(x)-x)^2}{g(x)+x-2f(x)}.$$
ii. Prove that if $x\in(0,c)$ then $h(x)\in(x,c)$.
iii. Analyze whether the sequence $\{a_n\}$ satisfying $a_1\in(0,c),a_{n+1}=h(a_n)$ for $n\in\mathbb N$ converges. Prove that the sequence $\{\tfrac{a_{n+1}-c}{a_n-c}\}$ converges and find its limit.
(c) Assume that the calculator approximates every number $b\in[-2,2]$ by number $\overline b$ having $p$ decimal digits after the decimal point. We are performing the following sequence of operations on that calculator:
1) Set $a=0.72$;
2) Calculate $\delta(a)=\overline{f(a)}-a$;
3) If $|\delta(a)|>0.5\cdot10^{-p}$, then calculate $\overline{h(a)}$ and go to the operation $2)$ using $\overline{h(a)}$ instead of $a$;
4) If $|\delta(a)|\le0.5\cdot10^{-p}$, finish the calculation.
Let $\overleftrightarrow c$ be the last of calculated values for $\overline{h(a)}$. Assuming that for each $x\in[0.72,0.73]$ we have $\left|\overline{f(x)}-f(x)\right|<\epsilon$, determine $\delta(\overleftrightarrow c)$, the accuracy (depending on $\epsilon$) of the approximation of $c$ with $\overleftrightarrow c$.
(d) Assume that the sequence $\{a_n\}$ satisfies $a_1=0.72$ and $a_{n+1}=f(a_n)$ for $n\in\mathbb N$. Find the smallest $n_0\in\mathbb N$, such that for every $n\ge n_0$ we have $|a_n-c|<10^{-6}$.
Let $t(A)$ denote the sum of elements of a nonempty set $A$ of integers, and define $t(\emptyset)=0$. Find a set $X$ of positive integers such that for every integers $k$ there is a unique ordered pair of disjoint subsets $(A_{k},B_{k})$ of $X$ such that $t(A_{k})-t(B_{k}) = k$.
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)
[i]Proposed by Bruno Le Floch, France[/i]